Problem: Simplify; express your answer in exponential form. Assume $x\neq 0, k\neq 0$. $\dfrac{{(x^{-1})^{3}}}{{(x^{5}k^{2})^{-5}}}$
Explanation: To start, try working on the numerator and the denominator independently. In the numerator, we have ${x^{-1}}$ to the exponent ${3}$ . Now ${-1 \times 3 = -3}$ , so ${(x^{-1})^{3} = x^{-3}}$ In the denominator, we can use the distributive property of exponents. ${(x^{5}k^{2})^{-5} = (x^{5})^{-5}(k^{2})^{-5}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(x^{-1})^{3}}}{{(x^{5}k^{2})^{-5}}} = \dfrac{{x^{-3}}}{{x^{-25}k^{-10}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{-3}}}{{x^{-25}k^{-10}}} = \dfrac{{x^{-3}}}{{x^{-25}}} \cdot \dfrac{{1}}{{k^{-10}}} = x^{{-3} - {(-25)}} \cdot k^{- {(-10)}} = x^{22}k^{10}$.